Latent variable modeling in Multivariate analysis

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Latent variable modeling in Multivariate analysis

• -A method to find useful information from raw
  data.
• Zheng LI
• Michigan State University

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Overview

• 1.Multivariate analysis
• 2.Latent variable modeling
• 3.Advance in Latent variable modeling
• 4.Application of Latent variable modeling

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What does multivariate mean?

• More than one variable are needed to describe a
  system
• For example
• Weather
• 1.Humidity
• 2.Temperature
• 3.Pressure
• And so on

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What can (do we want) multivariate analysis
(to)do?

• 1.Find an easier way to represent data.
• 2.Find out the distribution of the data.
• 3.Develop a model to predict a property of
  interest based on the other descriptors.
• For example
• Possibility of rainingF(humidity,
  temperature, pressure,)
• 4.Find out the cause-effects relation from the
  data.

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Difficulties in the way!

• How to deal with high dimensional system?
• Sometimes the dimensions ( number of
  variables) even exceeds the number of samples.
• How to deal with a system whose variables are not
  independent?
• Latent variable modeling is the solution!!!!

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Why latent Variables?

• Latent variables are independent to each other
  and more appropriate to be used to represent
  system instead of the original variables.
• Reduce the number of the variables, simplify the
  investigated system.

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Assumption of Latent variable modeling

• The investigated system is influenced by just a
  few underlying variables.

measurements
x1 x2 x3 x4 x5 x6 x7
Linear or nonlinear combination
Latent variable
Lx1 Lx2 Lx3
estimate
Sx1 Sx2 Sx3
system
measure
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Assumption of Latent variable models (continue)

• Lx s are estimates of Sx s, and they can span
• the same space. And S is related to L by usually
  an unknown rotation matrix R.

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How to define latent variables

• PCA principal component analysis
• PLS- partial least squares
• FDA-Fisher discriminant analysis
• are three common methods used to derive latent
  variables.

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PCA
 
Eigenvalue decomposition of X
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PLS
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FDA
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FDA
Eigenvalue decomposition of w-1B
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How to decide the number of latent variables?

• In PCA and FDA
• Based on the percent of the variance being
  explained by the latent variable model.

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How to decide the number of latent variables?
(continue)

• In PLS
• F-test

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Advance in Latent variable model research

• 1.Multiblock methods.
• 2.Nonlinear methods.
• 3.Application in causality relation discovery.

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Multiblock methods

• Multiblock means separate variables into
  different groups according to expert knowledge or
  variable clustering.
• Including MBPCA and MBPLS

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Latent variable in Causal modeling

• Latent variable modeling are used to identify the
  important factors in a system based on the
  analysis of loadings ( the coefficients of the
  latent variables).

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Latent variable in Causal modeling(continue)
Latent variable
Bayesian network analysis
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Application in metabolic flux analysis (MFA)
measurements
MFA MODEL
Flux value data
Multivariate analysis
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Application in metabolic flux analysis (MFA)

• Objective
• To develop model to relate metabolic fluxes, then
  identify fluxes that is important to the function
  of a cell.
• Urea productionF(other flux value of liver cell)

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Application in metabolic flux analysis-FDA
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Application in metabolic flux analysis- PLS
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Application in metabolic flux analysis- PLS
(continue)
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Application in metabolic flux analysis- causal
modeling
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Conclusion

• Latent variable modeling can help us to simplify
  a system.
• Latent variable modeling can help us to uncover
  the underlying mechanism of the system.